The optical design community has progressed in parallel with the advancements of computer aided design (CAD) and computer aided geometry design (CAGD) for many years. As computer aided design has progressed from simple curves, conics, and aspheric polynomials the applications have transitioned to optical design shortly thereafter or in parallel. Many problems in computer aided design allow for error many orders of magnitude higher than those tolerances allowed in optical design. The optical design community following its own relentless pursuit of diffraction limited performance and has pushed transversely to improve computer aided design to higher levels of precision. Gradually, more difficult optical problems have required creativity to push computer aided design further into the realm of optical design code. There are typically resistances to do so as the mathematics concerning simple curves, and aspherics, the associated wavefronts and aberrations is well understood. Metrology to inspect such advanced geometries also has to come up to speed in parallel, otherwise design realized on the computer cannot be verified with empirical data to ensure that what was designed was indeed manufactured precisely.
Many designers were the most resistant to embracing the computer-aided design world largely due to limitations in the free-form design space. The entertainment and defense industries with vast monetary resources has pushed the realization of more general design tools to the computer design space. The parametric mathematics has progressed from Bernstein polynomials, to Bezier curves, B-splines, and then tensor product surfaces or NURBS (Non-Uniform Rational B-Splines). Computer aided geometry design today is largely dominated by NURBS and sub-division surface modeling. In an effort to advance local refinement and free-form deformation T-splines are introduced as a new generalization of NURBS. To fully appreciate the advancement it is necessary to compile comparisons between the two topologies, the mechanics of knot-insertion/deletion, and how the non-imaging and imaging community may become beneficiaries of the application.
In the optical design arts for example, it would be desirable to progress from simple spherical control surfaces to conics and aspherics before entering the t-spline lens space. In some applications a merit function can be achieved with sequences of simple spherical surfaces. In many non-imaging applications which require tailoring the light emanating from an extended source spherical surfaces offer insufficient optical control. Progressing to a NURBS surface or spline patch grid surface presents challenges.
Control of the light at a local control point can be performed by iterative Cartesian oval calculations, but it is a tedious process with so many superfluous control points in close proximity to a local surface change. If one desires to add control perturbation to a section in-between the surface spline control points knot insertion produces exponentially greater complexity. Trying to add one knot at an optical control surface section of interest requires adding an entire row of control points to satisfy the rectangular NURBS grid topology.
One way to locally refine is to trim the NURBS with a cutting surface and then add control points. Joining additional surfaces to the trimmed NURBS becomes a problem and rips can frequently occur. For some models the rips and discontinuities at a NURBS junction can be ignored due to coarse machine tolerances. Rips and gaps however, produce problems for efficient and accurate optical raytracing, requiring post-process healing, and iterative surface approximation routines. These rips are more likely to occur when attempting to merge bezier edge curves with dissimilar continuity.
For the foregoing reasons, there is a need in the optical design industry to develop a method to refine geometries without the addition of superfluous control points.
There is also a further need to develop a method for interoperability between NURBS and T-splines surfaces for optical design that will accelerate adaptation allowing for compatibility with opto-mechanical software, tooling, and manufacturing.